# oc1S: Operating Characteristics for 1 Sample Design In RBesT: R Bayesian Evidence Synthesis Tools

## Description

The `oc1S` function defines a 1 sample design (prior, sample size, decision function) for the calculation of the frequency at which the decision is evaluated to 1 conditional on assuming known parameters. A function is returned which performs the actual operating characteristics calculations.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```oc1S(prior, n, decision, ...) ## S3 method for class 'betaMix' oc1S(prior, n, decision, ...) ## S3 method for class 'normMix' oc1S(prior, n, decision, sigma, eps = 1e-06, ...) ## S3 method for class 'gammaMix' oc1S(prior, n, decision, eps = 1e-06, ...) ```

## Arguments

 `prior` Prior for analysis. `n` Sample size for the experiment. `decision` One-sample decision function to use; see `decision1S`. `...` Optional arguments. `sigma` The fixed reference scale. If left unspecified, the default reference scale of the prior is assumed. `eps` Support of random variables are determined as the interval covering `1-eps` probability mass. Defaults to 10^{-6}.

## Details

The `oc1S` function defines a 1 sample design and returns a function which calculates its operating characteristics. This is the frequency with which the decision function is evaluated to 1 under the assumption of a given true distribution of the data defined by a known parameter θ. The 1 sample design is defined by the prior, the sample size and the decision function, D(y). These uniquely define the decision boundary, see `decision1S_boundary`.

When calling the `oc1S` function, then internally the critical value y_c (using `decision1S_boundary`) is calculated and a function is returns which can be used to calculated the desired frequency which is evaluated as

F(y_c|θ).

## Value

Returns a function with one argument `theta` which calculates the frequency at which the decision function is evaluated to 1 for the defined 1 sample design. Note that the returned function takes vectors arguments.

## Methods (by class)

• `betaMix`: Applies for binomial model with a mixture beta prior. The calculations use exact expressions.

• `normMix`: Applies for the normal model with known standard deviation σ and a normal mixture prior for the mean. As a consequence from the assumption of a known standard deviation, the calculation discards sampling uncertainty of the second moment. The function `oc1S` has an extra argument `eps` (defaults to 10^{-6}). The critical value y_c is searched in the region of probability mass `1-eps` for y.

• `gammaMix`: Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function `oc1S` takes an extra argument `eps` (defaults to 10^{-6}) which determines the region of probability mass `1-eps` where the boundary is searched for y.

Other design1S: `decision1S_boundary()`, `decision1S()`, `pos1S()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53``` ```# non-inferiority example using normal approximation of log-hazard # ratio, see ?decision1S for all details s <- 2 flat_prior <- mixnorm(c(1,0,100), sigma=s) nL <- 233 theta_ni <- 0.4 theta_a <- 0 alpha <- 0.05 beta <- 0.2 za <- qnorm(1-alpha) zb <- qnorm(1-beta) n1 <- round( (s * (za + zb)/(theta_ni - theta_a))^2 ) theta_c <- theta_ni - za * s / sqrt(n1) # standard NI design decA <- decision1S(1 - alpha, theta_ni, lower.tail=TRUE) # double criterion design # statistical significance (like NI design) dec1 <- decision1S(1-alpha, theta_ni, lower.tail=TRUE) # require mean to be at least as good as theta_c dec2 <- decision1S(0.5, theta_c, lower.tail=TRUE) # combination decComb <- decision1S(c(1-alpha, 0.5), c(theta_ni, theta_c), lower.tail=TRUE) theta_eval <- c(theta_a, theta_c, theta_ni) # evaluate different designs at two sample sizes designA_n1 <- oc1S(flat_prior, n1, decA) designA_nL <- oc1S(flat_prior, nL, decA) designC_n1 <- oc1S(flat_prior, n1, decComb) designC_nL <- oc1S(flat_prior, nL, decComb) # evaluate designs at the key log-HR of positive treatment (HR<1), # the indecision point and the NI margin designA_n1(theta_eval) designA_nL(theta_eval) designC_n1(theta_eval) designC_nL(theta_eval) # to understand further the dual criterion design it is useful to # evaluate the criterions separatley: # statistical significance criterion to warrant NI... designC1_nL <- oc1S(flat_prior, nL, dec1) # ... or the clinically determined indifference point designC2_nL <- oc1S(flat_prior, nL, dec2) designC1_nL(theta_eval) designC2_nL(theta_eval) # see also ?decision1S_boundary to see which of the two criterions # will drive the decision ```